Optimal. Leaf size=118 \[ \frac{2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^2 (p+2)} \]
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Rubi [A] time = 0.0913054, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {1652, 444, 43, 12, 365, 364} \[ \frac{2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^2 (p+2)} \]
Antiderivative was successfully verified.
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Rule 1652
Rule 444
Rule 43
Rule 12
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx &=\int 2 d e x^2 \left (d^2-e^2 x^2\right )^p \, dx+\int x \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (d^2-e^2 x\right )^p \left (d^2+e^2 x\right ) \, dx,x,x^2\right )+(2 d e) \int x^2 \left (d^2-e^2 x^2\right )^p \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (2 d^2 \left (d^2-e^2 x\right )^p-\left (d^2-e^2 x\right )^{1+p}\right ) \, dx,x,x^2\right )+\left (2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^2 (1+p)}+\frac{\left (d^2-e^2 x^2\right )^{2+p}}{2 e^2 (2+p)}+\frac{2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0813278, size = 110, normalized size = 0.93 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (4 d e^3 x^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{3 \left (d^2-e^2 x^2\right ) \left (d^2 (p+3)+e^2 (p+1) x^2\right )}{(p+1) (p+2)}\right )}{6 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.666, size = 0, normalized size = 0. \begin{align*} \int x \left ( ex+d \right ) ^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.59443, size = 440, normalized size = 3.73 \begin{align*} d^{2} \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{2 d d^{2 p} e x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} + e^{2} \left (\begin{cases} \frac{x^{4} \left (d^{2}\right )^{p}}{4} & \text{for}\: e = 0 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{2 e^{4}} - \frac{x^{2}}{2 e^{2}} & \text{for}\: p = -1 \\- \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac{d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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